×

Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise. (English) Zbl 1209.60038

Summary: We study the asymptotic behavior of solutions to the stochastic sine-Gordon lattice equations with multiplicative white noise. We first prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arnold, L., Random dynamical systems, (1998), Springer Berlin
[2] Bates, P.W.; Lisei, H.; Lu, K., Attractors for stochastic lattice dynamical systems, Stoch. dyn., 6, 1-21, (2006) · Zbl 1105.60041
[3] Bates, P.W.; Lu, K.; Wang, B., Attractors for lattice dynamical systems, Internat. J. bifur. chaos, 11, 143-153, (2001) · Zbl 1091.37515
[4] Beyn, W.-J.; Pilyugin, S.Yu., Attractors of reaction diffusion systems on infinite lattices, J. dynam. differential equations, 15, 485-515, (2003) · Zbl 1041.37040
[5] Cahn, J.W., Theory of crystal growth and interface motion in crystalline materials, Acta metall., 8, 554-562, (1960)
[6] Caraballo, T.; Lu, K.N., Attractors of stochastic lattice dynamical systems with a multiplicative noise, Front. math. China, 3, 317-335, (2008) · Zbl 1155.60324
[7] Chow, S.-N.; Mallet-Paret, J., Pattern formulation and spatial chaos in lattice dynamical systems, IEEE trans. circuits syst., 42, 746-751, (1995)
[8] Chow, S.-N.; Mallet-Paret, J.; Shen, W., Traveling waves in lattice dynamical systems, J. differential equations, 149, 248-291, (1998) · Zbl 0911.34050
[9] Chua, L.O.; Roska, T., The CNN paradigm, IEEE trans. circuits syst., 40, 147-156, (1993) · Zbl 0800.92041
[10] Chua, L.O.; Yang, L., Cellular neural networds: theory, IEEE trans. circuits syst., 35, 1257-1272, (1988) · Zbl 0663.94022
[11] Fan, X., Random attractor for a damped sine-Gordon equation with white noise, Pacific J. math., 216, 63-76, (2004) · Zbl 1065.37057
[12] Fan, X., Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. anal. appl., 24, 767-793, (2006) · Zbl 1103.37053
[13] Han, X.; Shen, W.; Zhou, S., Random attractors for stochastic lattice dynamical systems in weighted spaces, J. differential equations, 250, 1235-1266, (2011) · Zbl 1208.60063
[14] Lv, Y.; Sun, J.H., Dynamical behavior for stochastic lattice systems, Chaos solitons fractals, 27, 1080-1090, (2006) · Zbl 1134.37350
[15] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (2007), Springer-Verlag New York · Zbl 0516.47023
[16] Shen, W., Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. appl. math., 56, 1379-1399, (1996) · Zbl 0868.58059
[17] Wang, X.; Li, S.; Xu, D., Random attractors for second-order stochastic lattice dynamical systems, Nonlinear anal., 72, 483-494, (2010) · Zbl 1181.60103
[18] Zhou, S., Attractors for second order lattice dynamical systems, J. differential equations, 179, 605-624, (2002) · Zbl 1002.37040
[19] Zhao, C.; Zhou, S., Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. math. anal. appl., 354, 78-95, (2009) · Zbl 1192.37106
[20] Zinner, B., Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. math. anal., 22, 1016-1020, (1991) · Zbl 0739.34060
[21] Zinner, B., Existence of traveling wavefront solutions for the discrete Nagumo equation, J. differential equations, 96, 1-27, (1992) · Zbl 0752.34007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.